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A114581 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k UDH's starting at level 0 (U=(1,1),H=(1,0),D=(1,-1)).

%I A114581 #3 Mar 30 2012 17:36:07
%S A114581 1,1,2,3,1,7,2,16,5,40,10,1,100,24,3,256,58,9,663,149,22,1,1741,386,
%T A114581 57,4,4620,1017,147,14,12376,2702,392,40,1,33416,7248,1053,113,5,
%U A114581 90853,19590,2859,312,20,248515,53318,7803,870,65,1,683429,145984,21420,2428
%N A114581 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n having k UDH's starting at level 0 (U=(1,1),H=(1,0),D=(1,-1)).
%C A114581 Row n contains 1+floor(n/3) terms. Row sums are the Motzkin numbers (A001006). Column 0 yields A114582. Sum(k*T(n,k),k=0..floor(n/3))=A002026(n-2).
%F A114581 G.f.=2/[1-z-2tz^3+2z^3+sqrt(1-2z-3z^2)].
%e A114581 T(7,2)=3 because we have (UDH)(UDH)H, H(UDH)(UDH) and (UDH)H(UDH), where U=(1,1),H=(1,0),D=(1,-1) (the UDH's starting at level 0 are shown between parentheses).
%e A114581 Triangle starts:
%e A114581 1;
%e A114581 1;
%e A114581 2;
%e A114581 3,1;
%e A114581 7,2;
%e A114581 16,5;
%e A114581 40,10,1;
%p A114581 G:=2/(1-z-2*t*z^3+2*z^3+sqrt(1-2*z-3*z^2)): Gser:=simplify(series(G,z=0,21)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 17 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/3)) od; # yields sequence in triangular form
%Y A114581 Cf. A001006, A114582, A002026.
%K A114581 nonn,tabf
%O A114581 0,3
%A A114581 _Emeric Deutsch_, Dec 09 2005